A GPT must specify what kind of physical systems one can find in the lab, as well as rules to compute the outcome statistics of any experiment involving labeled preparations, transformations and measurements.
[3] Notably, a small set of physically motivated axioms is enough to single out the GPT representation of quantum theory.
[4][5][6][7] The mathematical formalism of GPTs has been developed since the 1950s and 1960s by many authors, and rediscovered independently several times.
The earliest ideas are due to Segal[8] and Mackey,[9] although the first comprehensive and mathematically rigorous treatment can be traced back to the work of Ludwig, Dähn, and Stolz, all three based at the University of Marburg.
[10][11][12][13][14][15] While the formalism in these earlier works is less similar to the modern one, already in the early 1970s the ideas of the Marburg school had matured and the notation had developed towards the modern usage, thanks also to the independent contribution of Davies and Lewis.
[16][17] The books by Ludwig and the proceedings of a conference held in Marburg in 1973 offer a comprehensive account of these early developments.
[18][4] The term "generalized probabilistic theory" itself was coined by Jonathan Barrett in 2007,[19] based on the version of the framework introduced by Lucien Hardy.
Additionally it is always assumed that measurement outcomes and physical operations are affine maps, i.e. that if
Note that physical operations are a subset of all affine maps which transform states into states as we must require that a physical operation yields a valid state even when it is applied to a part of a system (the notion of "part" is subtle: it is specified by explaining how different system types compose and how the global parameters of the composite system are affected by local operations).
For practical reasons it is often assumed that a general GPT is embedded in a finite-dimensional vector space, although infinite-dimensional formulations exist.
[21][22] Classical theory is a GPT where states correspond to probability distributions and both measurements and physical operations are stochastic maps.
Standard quantum information theory is a GPT where system types are described by a natural number
Systems compose via the tensor product of the underlying complex Hilbert spaces.
[19] Other examples are theories with third-order interference[24] and the family of GPTs known as generalized bits.